Parallels Between Constructing Dynamic Figures and Constructing Computer Programs.

Constructing dynamic figures is an activity central to dynamic geometry systems and requires a methodology that augments the traditional approach to construction in geometry. Teachers and students who use these systems need exposure to such a methodology for constructing and exploring dynamic figures. This article advocates that constructing dynamic figures involves constructing programs for dynamic geometry systems and one such system GDRev is used to implement dynamic figure construction programs. A nontrivial construction problem is analyzed using a combination of procedural and declarative approaches to dynamic figure construction and the analysis is used to illustrate a three-step methodology for constructing dynamic figures consisting of problem requirements, formal specifications, and implementation.

The construction of dynamic figures is a new activity provided by recently developed dynamic geometry systems such as Cabri Geometry (Laborde, & Bellermain 1994), Geometer's Sketchpad (Jackiw, 1995), and Geospecif (Allen, Idt, & Trilling, 1993). A publication of the Mathematical Association of America (King & Schattschneider, 1997) illustrated how abundantly these dynamic geometry systems are utilized in teaching at all levels. Constructing dynamic figures requires a methodology that augments the traditional approach to construction in geometry and teachers and students who use these systems need a methodology for constructing and exploring dynamic figures. This article advocates the use of dynamic figures for solving construction problems in geometry and shows that constructing dynamic figures involves constructing programs for dynamic geometry systems. One such system GDRev (an acronym for the French equivalent of Reversible Dynamic Geometry) (Channac, 1998) is discussed and is used to implement dynamic fi gure construction programs.

The presentation begins with a nontrivial construction problem and proceeds to explore the problem using a combination of procedural and declarative approaches to dynamic figure construction as embodied in one or more of the four dynamic geometry systems mentioned in the preceding paragraph. The exploration process itself leads to a better understanding of what is a dynamic figure and of what is the difference between a dynamic figure and its geometric component. A three-step methodology for constructing dynamic figures is followed and leads to the construction of a dynamic figure that solves the initial problem. The three steps in the methodology are: (a) problem requirements, (b) formal specification, and (c) implementation. These steps embody the computer program development process; making them explicit in the teaching process is both a way to help students change how they approach problem solving in geometry and a way to interrelate geometry and programming.

The central problem for this article arises from a question asked by a former president of the National Council of Teachers of Mathematics, Frank Allen, to Joe Cieply. The problem was sent in a message to Cabri-Forum by Cieply and in the message he said that he had worked on the problem off and on in Cabri and in Sketchpad and that he had not generated the insight he needed to solve the problem. The problem can be stated simply: Given an angle A, and medians BD and CE, construct triangle ABC with BD and CE as its medians. Cieply went on to say in the message that he had used all the trace, animation, and so forth, tools these systems provide to try to get an insight into solving the problem but so far had not solved the problem.

It is important to note that this message does not ask specifically for a solution, but for a way to guess a solution by using the tools for doing dynamic geometry provided by two well-known systems. Following an attitude similar to the one in the message, the purpose here is not only to give a solution to this problem but, more importantly, to present a general methodology for solving problems whose solutions require the construction of dynamic figures. It is worthwhile to note that this message is formulated in the typical way that one poses problems about the construction of dynamic figures; namely, the message alludes to informal ideas about the sought after figures, then to using these ideas to better specify what you want, and, finally, to finding effective implementations for a figure. As stated here, this three-stage process mirrors the program development cycle of requirements, specification, followed by implementation.

The mere formulation of a problem is often far more essential than its solution, which may be a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science.

Attributed to Albert Einstein 

The main ideas to be developed in this article are the following:

1. Constructing a dynamic figure is like designing a computer program in that such activity requires the design of an algorithm to be implemented on a dynamic geometry system; such activity is traditionally considered a computer science activity. This point of view thus connects dynamic figure construction to powerful ideas and methods outside of mathematics. Furthermore, the heavy use of declarative information in the methodology shows how problems that are essentially teaching issues provide computer science with new areas of research

2. Designing a computer program occurs traditionally in three steps: (a) establish an informal description of the problem (requirements); (b) determine a formal (logical) specification of the outcome of the program; and (c) implement the specification in a specific language. This methodology, which is taught to computer science students, is typical of the modern no-fault approach for constructing programs. The attitude behind using such an approach to problem solving often seems in contrast to the atmosphere sometimes found among the users of dynamic geometry systems, an attitude that supports unordered, nonsystematic draggings of a drawing on a screen. In fact, the attitude should be quite different; a student should know before using a dynamic geometry system what the requirements are and the specification of a dynamic figure to be constructed. It is possible that current expert users of dynamic geometry systems may have started using these systems already knowing too much geometry and already having too hi ghly honed intuition regarding the methodology for constructing dynamic figures. What is natural for the current experts may not be the case for beginning students. In any case, it seems worthwhile that students (of both computer programming and of dynamic geometry) spend time thinking about a specification before starting to implement in a specific language (programming language or dynamic geometry system).

3. Manipulating declarative information, as found in the logical specifications of geometric figures and computer programs, provides a more general context for finding, modifying, and implementing solutions to problems. Working from declarative information provides a powerful cognitive tool that is often underutilized or neglected altogether by those who restrict themselves exclusively to the use of procedural dynamic geometry systems like Cabri and Sketchpad. The GDRev system provides an interface that allows construction of dynamic figures and, simultaneously, produces the specification for the figure being constructed. In fact, at any time during construction, the interface allows direct manipulation of the specification, including modifying the specification declaratively and redrawing the figure conforming to the modified specification. The system is able to reconstruct dynamic figures from a dynamic specification; that is, the specification associated with the geometric properties of the figure plus cer tain designated points in the figure.

The remainder of the article is organized as follows. The three-step methodology for constructing a dynamic figure is presented through the construction problem presented previously. In particular, the loci method is introduced and exploited to obtain the problem requirements, which in turn leads to a formal specification of two dynamic figures. Several attempts to implement these dynamic figures directly from their specifications are given. GDRev is used for the implementations and the system interface is described along with an explanation of some of its major functions. In describing the successive implementation attempts, the mathematical (Colmerauer & Benhamou, 1993) and (Bouhineau, 1996) technical issues that affect successful implementations are discussed. These implementation attempts make heavy use of declarative information provided by and to the system. Along the way what specific geometric knowledge is necessary to find a solution is seen. At the end how to obtain procedurally an optimized dynami c figure using Cabri is shown. The article concludes with observations on the connections between dynamic figure construction and both the methodology presented in this work and the use of declarative geometric programming.

METHODOLOGY FOR BUILDING DYNAMIC FIGURES

Recall the problem from section 1: Given an angle A, and medians BD and CE, construct triangle ABC with BD and CE as its medians. Also, recall that the purpose here is not just to give a solution to this problem, but to present a general methodology to attack such problems by constructing dynamic figures. The approach to the problem will be to divide it into two subproblems and to solve each of them using what is traditionally called the method of loci. The construction of loci intrinsically calls for the construction of a dynamic figure, whether carried out in the human imagination or with the aid of external tools, and the advent of dynamic geometry systems with their capacity to construct loci automatically has given users a concrete tool with which they can experimentally construct and modify loci efficiently and effectively. In fact, for many students, use of the locus tool found in these systems has made teaching the loci method in school accessible.

The first issue confronting students as they read the statement of a problem such as this one is that of being able to interpret what this problem represents with respect to the knowledge they have from their school geometry course. Yes, they know what an angle is, what a triangle is, and what are medians of triangles. But they think of them in terms of specific drawings; they don't know how to think of or describe an arbitrary angle or median on paper or on a computer screen let alone in their minds. Here is where some decisions about problem representation must be made; here is where the student must come up with a way to model an arbitrary angle and arbitrary medians. Here is where the student needs to decide whether to use three noncolinear points, two intersecting segments, or two intersecting lines, to represent an angle, and where to place them. The student needs to think of medians as segments having endpoints, length, direction, and placement.

Statements and Requirements of Two Subproblems

Note that the original problem constrains the angle A to be equal to a given angle and the medians BD and CE to be fixed. The key to defining the subproblems is to relax one or more of these constraints. To obtain subproblem 1 the constraint that angle A be equal to a given fixed angle is removed. To obtain subproblem 2 constraints on the length and direction of median CE are removed.

The solution to the original problem is given by the intersection of the two loci because the intersection point is positioned so that the median CE has the required given length and direction and so that angle A of triangle ABC is equal to angle A' of the given triangle A'B'C'. Together this provides a triangle ABC whose medians BD and CE are the given ones and whose angle A is the one sought after. In effect, drawings 1 and 2 provide geometric models of the assumptions and the conclusions of the two subproblems. However, the real issue is the development of a systematic method that starts with the assumptions of the subproblems, develops specifications for constructing dynamic figures that will produce the required triangles and loci, and finally provides an effective implementation of these dynamic figures. As noted at the end of the introduction, heavy use will be made of declarative information both to develop the specification and the implementation.

Specifications

For each subproblem the approach will be to construct, using the dynamic geometry system GDRev, a geometric figure representing the triangle ABC and respecting the constraints of the subproblem. GDRev generates a logical specification for the figure as it is being constructed. This specification will be used to generate a dynamic figure which also respects the constraints of the subproblem and which will generate the sought after locus as the direction I rotates through 3600.

Implementations

For subproblem 1, place points A,B,C,C' in the plane as suggested in drawing 3. Construct triangle ABC by constructing segments AB,BC,AC. Then construct the medians BD and CE. Construct the line through C' and parallel to the segment CE. Use the compass tool to construct the circle with center at C' and with radius equal to the length of the segment CE. Let E' be the intersection of the circle and the line through C'. Construct a halfline through C' and E' and let M be an arbitrary point on this half-line. Let E" be a point on the circle and construct the segment C'E". In drawing 3 observe that the names of the points A,B,C,C',E" are all underlined. In this drawing the underlined points indicate those points from which the other geometric objects in the figure are constructed. What is more important is that, if the figure is drawn with a dynamic geometry system, each of those five points can be dragged and the entire figure is redrawn and all geometric properties of the original figure are preserved. Such a set of points for a figure is called a set of base points; often each point in the set is called fixed while other points in the figure are called free. Note that distance d and direction I in drawing 1 are represented in drawing 3, respectively, by the length of C'E" and by the direction of the half-line through C' and M.

The GDRev system provides an interactive interface with pull-down menu bars and two main windows. One window gives the user a graphics interface to construct and edit drawings of geometric figures similar to the way the Cabri and Sketchpad interfaces do (see drawings 3 & 4). Another window displays the corresponding logical specification for a figure being constructed and allows the user to edit the specification as well. The specification window has several tab boxes that allow access to the figure's geometric specification, to the figure's current dynamic specification, to an editor of the geometric specification, and to a procedural construction for the current geometric figure (if there is one). Diagram 1 provides a view of the principal features of the GDRev interface.

A dynamic figure consists of a geometric specification for the figure plus a set of base points. Consequently, the dynamic figure associated with drawing 3 is the preceding specification plus the set of base points: A, B, C, C', E". This dynamic figure will not produce the solution to subproblem 1, in part, because we cannot drag on the direction represented by the line through C'M. We need the dynamic figure that produces the figure illustrated in drawing 4; namely, the same specification as previously, but with base points: B, D, C', E", M. The GDRev interface has a pull-down menu that provides tools that interact with the figure's dynamic specification to free (i.e., unfix) the base points A, C and, subsequently, to fix the points D, M. GDRev then redraws the figure preserving all the original geometric properties but with B, D, C', E", M as the base points capable of being dragged (see drawing 4). Using the trace/locus tool and dragging on the point M in a circular manner produce the desired locus seen i n drawing 1. Note the importance of the specification in this process; the specification is declarative information that determines the figure independent of any particular drawing of the figure. Furthermore, the specification is also independent of the order of construction of the objects in any drawing. An important geometric object in the specification is the half-line and an important property in the specification is the one that says that the half-line passes through the point E'. This combination in the specification assures that E' is reconstructed on the correct side of the circle each time the dynamic figures moves.

Subproblem 2

The approach to subproblem 2 proceeds in a manner similar to that taken above for subproblem 1. Place points A, B, C, B', C' in the plane as seen in drawing 5. Construct triangle ABC by constructing segments AB, BC, AC. Then construct the median BD. Construct the line through B' parallel to AR, the line through C' parallel to AC, and the point A" as the intersection of these two lines. Next construct the circle through the three points A", B', C'. Let A' be any point on the circle and M be a point on the line through B'A". This series of construction steps is illustrated by drawing 5 and produces a dynamic figure with base points A, B, C, B', C'. Note that the three equal angles, BAC, B'A"C', B'A'C' are all equal in measurement. Note that the direction 1 and the angle at A' in drawing 2 are represented in drawing 5, respectively, by the direction of the line B 'M and the angle at A'. The key to this construction is the intermediate angle B'A"C' equal to both the original and the sought after angles and with sides parallel to the sides of the original angle.

The dynamic figure associated with drawing 5 (figure specification plus five base points, A, B, C, B', C') will not produce the solution to subproblem 2, in part, because we cannot drag on the direction represented by the line through B'M. We need the dynamic figure that produces drawing 6; namely, we need the same specification but with base points: B, D, A', B', C', M. As was done for subproblem 1, the GDRev interface allows the user to interact with the figure's dynamic specification to free (i.e., unfix) the base points A and C and, subsequently, to fix the points D, A', M. GDRev then redraws the figure preserving all the original geometric properties but with B, D, A', B', C', M as the base points (see drawing 6). Using the trace/locus tool and dragging on the point M in a circular manner produce the desired locus seen in drawing 2. This implementation works just fine as long as the line B'M intersects the circle in a point A" that lies above the segment B'A'. What changes are needed in the implementatio n if the line B'M intersects below the segment B'A'?

Methodology

This section began with an attempt to understand the requirements of the problem contained in Cieply's message by reformulating it into two subproblems. The key to this reformulation lay in relaxing constraints on lengths, directions, and angles in the original problem (drawings 1 & 2) and then modeling the subproblem requirements by constructing dynamic figures (3 & 5). The GDRev system interface is used to construct the figures and to obtain the specifications, which are part of them. Using the specifications developed for each of the subproblems, different dynamic figures (see drawings 4 & 6) for each of the subproblems are defined and implemented using the specific language provided by GDRev. This section illustrated in detail the cycle of requirements, specification, and implementation of dynamic figures used to solve the problem in the Cieply message.

RELATED IMPLEMENTATION ISSUES 

This section is devoted to specific declarative and procedural implementations of dynamic figures related to the ones presented in the previous section. GDRev provides us with implementations of the dynamic figures represented in drawings 3, 4, 5, 6. As mentioned previously GDRev dynamically and interactively creates a specification for a figure and reconstructs and modifies a figure using its specification and a set of base points, thus making it a system that functions declaratively. In fact, the GDRev interface provides an editor to modify a specification directly, including the base points, and then redraws the resulting dynamic figure.

Linear Implementations

In dynamic geometry systems, all points in a figure must be represented as pixels on a computer screen. What this is equivalent to mathematically is saying that all the base points have rational numbers as coordinates. However, coordinates of points constructed subsequently may not have rational coordinates and, thus, may not have an exact numerical representation as rational numbers. This situation makes placement of such points on the screen approximate. For example, this situation arises in calculating the coordinates of the points of intersection of two circles, of a line and a circle; the distance between two points with rational coordinates may not itself be rational. Special attention is sometimes given to implementations of dynamic figures that use only points and lines since these implementations produce exact placement of points and lines on computer screens and they are faster when dragging and reconstructing figures. Such implementations are referred to as linear because intersection points of lin es constructed using points with rational coordinates are solutions to systems of linear equations and, thus, have rational coordinates. Are there linear implementations for subproblems 1 and 2? Yes, and it raises an interesting issue for constructing dynamic figures automatically, namely, adding supplementary objects and properties to the specification of a geometric figure.

Linear implementation of subproblem 1. The methodology is the same as in the previous cases: use the problem requirements to obtain a specification. Place the points A, B, C, C', M in the plane. Construct the sides of the triangle AB, BC, AC, and the medians BD, CE along with their point of intersection G. Construct the segment CC' and the segment C'E' parallel to CE. Construct segments GG' and EE', each parallel to CC', and construct the line through E'M. Construct F' as the symmetric point of E' with respect to C' and the segment F'E" perpendicular to the line E'M. Construct F" as the symmetric point of E" with respect to C' and the segment F"E' perpendicular to the line E'M. Construct the segment G'G" parallel to the line through E'M. Note that E'E"F'F" is a rectangle and, thus, the segments C'E' and C'E" are equal. As before, GDRev provides a specification for the figure of drawing 7. Using the editor provided by the system, a well known geometric property of triangles is added to this specification, name ly, that the point G is two thirds of the distance from B to D. Also added to the specification is the property that G" is also two thirds of the distance from C' to E". The validity of this second property is clear from the parallelogram and rectangle constructed. This information is added to guarantee that the system has enough information to reconstruct the figure automatically using this linear implementation. It is an interesting mathematical exercise to ask students to explain why this is so.

Again, the dynamic figure obtained from the specification of the figure drawn in drawing 7 and the base points A, B, C, C', M will not produce the solution to subproblem 1. We need a dynamic figure with the points A and C free and with the point E" fixed. Use the GDRev interface to designate B, D, C', E", M as base points with the same specification to obtain the dynamic figure represented in drawing 8. Using the trace/locus tool and dragging on the point M in a circular manner produce the desired locus found in drawing 1. Again, note the importance of the specification in this process as well as the base points.

Linear implementation of subproblem 2. Place points A, B, C, B', C', and P in the plane as seen in drawing 9. Construct triangle ABC by constructing segments AB, BC, AC. Then construct the median BD and the segment B'P. Construct the line through B' parallel to AB, the line through C' parallel to AC, and the point M as the intersection of these two lines. Next construct the perpendicular bisectors of the segments B'C' and C'M and let O be their point of intersection. Construct the point S as the point symmetry point of B' with respect to O. Finally, construct the line through S perpendicular to B'P and call the intersection point A'. This series of construction steps is illustrated by drawing 9 and produces a dynamic figure with base points A, B, C, B', C', and P. Note that the three equal angles, BAC, B'MC', B'A'C' are all equal in measurement. Note that the direction 1 and the angle at A' in drawing 2 are represented in drawing 9, respectively, by the direction of the line B'M and the angle at A'. The key to this construction is the intermediate angle B'MC' equal to both the original and the sought after angles and with sides parallel to the sides of the original angle.

The dynamic figure associated with drawing 9 (figure specification plus six base points, A, B, C, B', C', P) will not produce the solution to sub-problem 2, in part, because we cannot drag on the direction represented by the line through B'M. We need the dynamic figure that produces drawing 10; namely, we need the same specification but with base points: B, D, A', B', C', M. As was done for sub-problem 1, the GDRev interface allows the user to interact with the figure's dynamic specification to free (i.e., unfix) the base points A, C, P and, subsequently, to fix the points D, A', M. GDRev then redraws the figure preserving all the original geometric properties but with B, D, A', B', C', M as the base points (see drawing 10). Using the trace/locus tool and dragging on the point M in a circular manner produce the desired locus seen in drawing 2. Note that in this implementation there are a number of intermediate points and lines and accompanying properties needed to be added to the specification of the origina l figure (drawing 2) in order to arrive at a linear implementation.

Procedural Implementation

In this section an entirely procedural implementation of a figure that provides a solution to the original problem is given. This implementation was inspired by the previous solutions to the two subproblems using the declarative approach to implementing dynamic figures. Such a procedural implementation can be carried out with systems like Cabri or Sketchpad. Drawing 11 shows the procedural construction of an optimized figure for discovering the locus of A. It uses a lot of specific geometric information, so much so, that a person who already knows how to use this information to obtain the figure might indeed already also know what the sought after locus looks like beforehand.

Drawing 11 is constructed using Cabri in the following way: Place the points B and G and the circle (O) in the plane. Construct D so that DG = 1/2 BG. C is chosen on (O) and E is constructed such that EG = 1/2 CG. Construct A at the intersection of the line through points B and E and the line through the points C and D. Then use the locus tool to construct the loci of A and E, respectively, as C moves around the circle (O). Both loci are circles. It can be shown easily that the locus of A can be obtained by a homothety of center B from the locus of B. The specification for the previous figure plus B, G, and (O) constitute a dynamic figure with the preceding procedural implementation.

CONCLUSION

The geometric information used in the declarative approach described in this article is basic and is taught in all elementary school geometry courses. By contrast, the information needed to make sense of the procedural implementation given in the Procedural Implementation section is not basic and this implementation would not even make sense to students without all of the discussion involved in creating the dynamic figures of drawings 2, 3, 4, and 5. Solving geometry construction problems using a declarative approach is often easier than using a procedural approach since it is often more straight forward and often uses more basic tools and elementary information.

Problem solving using a declarative dynamic geometry programming system to construct dynamic figures is important from a conceptual standpoint since it brings to the forefront a fruitful distinction between the geometric aspect and the dynamic aspect of dynamic geometry. It brings a sharper understanding of a dynamic figure as a combination of a specification and a set of base points. Every dragged drawing of a given dynamic figure still respects the specification and can only be dragged from the set of base points. More importantly, constructing dynamic figures employs the same three step methodology used in computer science to develop programs. A dynamic figure is a declarative geometry program, specification plus base points that accepts inputs and produces outputs when implemented on a declarative geometry processor like GDRev.

References

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Bouhineau, D. (1996). Solving geometrical constraint systems using CLP based on linear constraint solver. Proceedings of the International Conference on Artificial Intelligence and Symbolic Mathematics Computation, Steyr, Austria, September, 1996, Calmet, J., Cambell, J.A., & Pfalzgraf, J., (Eds.), Lecture Notes in Computer Science, vol. 1138, Springer-Verlag, pp. 274-288.

Channac, S. (1998). Un systeme cooperatif pour la resolution de contraintes geometriques. Actes des Septiemes Journees Francophones de Programmation Logique et Programmation par Contraintes, Nantes, France, Hermes, May 27-28, 1998, pp. 77-93.

Colmerauer, A., & Benhamou, F., (Eds.) (1993). Constraint logic programming, selected research. MIT Press.

Jackiw, N. (1995). The Geometer's Sketchpad. Berkeley, CA: Key Curriculum Press.

King, J., & Schattschneider, D. (Eds.) (1997). Geometry turned on! Dynamic software in learning, teaching, and research.' MAA Notes Series, vol. 41, Mathematical Association of America, 1997.

Laborde, J.-M., & Bellemain, F. (1994). Cabri Geometry II. Dallas, TX: Texas Instruments.